how-to-find-x-2048-x-2048-if-x-x-1-5-1-2-

Question Number 76431 by john santu last updated on 27/Dec/19

h o w t o f i n d x^{2048} + x^{- 2048}

i f x + x^{- 1} = \frac{\sqrt{5} + 1}{2}

Commented by mathmax by abdo last updated on 28/Dec/19

w e h a v e x + x^{- 1} = \frac{1 + \sqrt{5}}{2} \Rightarrow {(x + x^{- 1})}^{n} = {(\frac{1 + \sqrt{5}}{2})}^{n} \Rightarrow

\sum_{k = 0}^{n} C_{n}^{k} x^{k} {(x^{- 1})}^{n - k} = {(\frac{1 + \sqrt{5}}{2})}^{n} \Rightarrow

\sum_{k = 0}^{n} C_{n}^{k} x^{k} x^{k - n} = {(\frac{1 + \sqrt{5}}{2})}^{n} \Rightarrow \sum_{k = 0}^{n} C_{n}^{k} x^{2 k} = {(\frac{1 + \sqrt{5}}{2})}^{n} x^{n}

c h a n g e x b y \frac{1}{x} \Rightarrow \sum_{k = 0}^{n} x^{- 2 k} = {(\frac{1 + \sqrt{5}}{2})}^{n} x^{- n} \Rightarrow

\sum_{k = 0}^{n} C_{n}^{k} x^{2 k} + \sum_{k = 0}^{n} C_{n}^{k} x^{- 2 k} = {(\frac{1 + \sqrt{5}}{2})}^{n} (x^{n} + x^{- n}) \Rightarrow

x^{n} + x^{- n} = \frac{\sum_{k = 0}^{n} C_{n}^{k} x^{2 k} + \sum_{k = 0}^{n} C_{n}^{k} x^{- 2 k}}{{(\frac{1 + \sqrt{5}}{2})}^{n}} = \frac{{(x^{2} + 1)}^{n} + {(x^{- 2} + 1)}^{n}}{{(\frac{1 + \sqrt{5}}{2})}^{n}} \Rightarrow

x^{2048} + x^{- 2048} = \frac{{(1 + x^{2})}^{2048} + {(1 + x^{- 2})}^{2048}}{{(\frac{1 + \sqrt{5}}{2})}^{2048}}

Answered by Kunal12588 last updated on 27/Dec/19

x + x^{- 1} = \frac{\sqrt{5} + 1}{2}

x^{2} + 2 + x^{- 2} = \frac{6 + 2 \sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2}

x^{2} + x^{- 2} = \frac{\sqrt{5} - 1}{2}

x^{4} + x^{- 4} = \frac{6 - 2 \sqrt{5}}{4} - 2 = - \frac{1 + \sqrt{5}}{2}

x^{8} + x^{- 8} = \frac{\sqrt{5} - 1}{2}

x^{16} + x^{- 16} = - \frac{1 + \sqrt{5}}{2}

⋮

x^{2048} + x^{- 2048} = \frac{\sqrt{5} - 1}{2}

Commented by john santu last updated on 27/Dec/19

t h a n k s s i r

Commented by mr W last updated on 27/Dec/19

w h a t i f i t i s t o f i n d x^{2020} + x^{- 2020} ?

Commented by Kunal12588 last updated on 27/Dec/19

I c a n f i n d x

x = \frac{(1 + \sqrt{5}) \pm i \sqrt{2 \sqrt{5} (\sqrt{5} - 1)}}{4}

b u t h o w t o g e t x^{2020} a n d \frac{1}{x^{2020}} ?

Commented by Kunal12588 last updated on 27/Dec/19

f_{n} = x^{n} + x^{- n}

g i v e n : f_{1} = \frac{\sqrt{5} + 1}{2}

f i n d : f_{2048}, f_{2020} a n d f_{n}